Concepts and contrasts
Hermann Grassmann and Bernard Bolzano
pp. 117-128
in: Steve Russ, Jörg Liesen (eds), From Past to Future: Graßmann's Work in Context, Berlin, Springer, 2011Abstract
Both Bolzano and Grassmann worked intensively for much of their lives on mathematical concepts – both technical concepts within mathematics such as line, number, function, continuity, extension, and broader concepts about mathematics such as its nature and classification, and what should be understood by a "proof". In this paper we shall compare some of the ways in which they worked with concepts because, while strikingly different, both scholars were led, through their analysis and development of concepts, often in a philosophical context, into fruitful new mathematics which was decades ahead of its time. The case is all the more interesting because of the close contextual parallels between the men which throws their differing treatment of concepts into greater relief. Furthermore, for Grassmann one of the major driving metaconcepts of his thinking about, and within,mathematics appears to have been the very concept of contrast, or of opposition (Gegensatz).
